NET 2021 Physics - Vector analysis

The following question appeared in NET 2021 Physics question paper 

Question:

The volume integral   

 is over a region τ bounded by a surface Ʃ ( an infinitesimal area element being

 

 where        is the outward unit normal. If it changes to

 

 

When the vector     is changed to

 

Then 

 can be expressed as

 Option a) 

 

Option b ) b) 

Option c) 

c)  

Option d) 

Solution :

Given

 

 As per the given problem 

 changes to

Substituting for 

we get 

which  reduces  to

 

.                                                             

As curl gradient of a scalar function is zero

The above equation can be written as

Hence

Let

                                                    

then

                                                                                                                 …………… (1)

Now using the vector  identity

The first tem of R.H.S becomes zero as    

                                                                 

And curl gradient of a scalar function is zero. Therefore

Substituting  the above equation in equation 1.

Using divergence theorem

Substituting for

 and simplifying

 So option c is the answer. 

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